Thanks, Marta, I didn't know that Asturians had experience this problem. It has appeared in my own family. I wondered if there is a genetic propensity to goiter and located several articles, such as this one:
https://www.ncbi.nlm.nih.gov/pubmed/16054835
That article reported a a strong genetic component in euthyroid familial goiter.

I don't know if there is a genetic propensity, but I know that there are different types of goiter. In the particular case of endemic goiter, it is the result of an iodine poor diet.

The genetic goiter seems to be due to an enzymatic defect that results in the lack of iodation of the hormone tyroxine.

There are some substances that induce goiter (an excess of iodine also results in goiter). Some pharmaceuticals are among those substances. For example, the best known is the amiodarone (antiarrhythmic medication used to treat ventricular tachycardia or ventricular fibrillation) that contains high amounts of iodine.

The genetic and environmental interactions of goiter can be complex. I fixed Art's link - it had a trailing "]article" that caused it not to work. Given the historic difficulty of transporting seafood to the mountainous regions of Asturias, I'm not surprised to find goiter there. Goiter was also common n some locations in the EEUU that are far from the sea.

Pregnancy can result in iodine deficiency because the bodies of the fetus and mother interact to provide the fetus with whatever it needs for growth and development. The mother's bones will give up calcium too.

Yes, there are many recommendations about to take iodine supplements not only during pregnancy but also while breastfeeding.
In iodine-deficient regions it seems that supplementation of breastfeeding-mothers with iodine is more effective than direct supplementation of the infant in order to reduce infant iodine deficiency.
Anyway, it seems that the amount of iodine that must be take needs to be carefully controlled, otherwise other problems may arise.